The trivial starting points: U(n,1)=0 U(1,k)=k U(2,2)=6, as pointed out by ThemePark. U(2,3) has 0, 1, 2, 10, 11, 12, 20, 21, 22. 9 digits. 9!=362880 so we can't just bruteforce it by hand. So I'm going to see if I can work out what would make it not work. 1 and 2 can't be adjacent, as that woul...

That's true, but that's not consistent with the presumption that the oracles are always correct. If they were basing it on real-world information, there's no way they'd remain accurate. I don't see why it's impossible. Let's give an example of a simple scenario: There is a box, it contains 2 dice, ...

That reveals your own irrationality. You are categorically giving alternative explanations a probability of zero so that they will never be the more likely explanations. No, I accept that there's more likely explanations after witnessing the events, what I'm saying is that it's irrational to think ...

Given by the puzzle, the chance that your prior information is false is zero. If the puzzle involves me being forced to believe that: I'm no longer a rational agent. Of course I won't stop betting, I'm under mind control. If the puzzle requires me to be convinced of that: The puzzle is an impossibl...

I think people arguing over the "highly unlikely scenario demanding additional explanation" just aren't putting enough prior probability into their bayesian analysis. And I think that you're putting a 100% probability into your bayesian analysis, which is always inaccurate. If you have en...

Kingreaper, But there is a possible solution (aBcdEf) wherein E is true and A isn't. This proves that F is false, and means that aBcdEf is the only possible solution. Notice that f is false in that solution, so aren't you basically saying "f must be false because there is a con...

For the original puzzle: E is true, because if D is true then C is true, B is false and A is true. If D were true, then only B would be false ( A b CDEF ). If D were false then only B and E would be true (a B cd E f). Both of these are consistent, until you look at F again; F says th...

I understand this portion of the solution. However, this only works if the participants are not 'perfect' logicians and/or cannot see each other at all times. We are told they can. Not at all, it works fine with them all seeing each other, as long as they can't read each other's minds. This being t...

I'm not familiar with this form of long multiplication, so I may make some incorrect assumptions. I'm working on the basis that we're multiplying by each of the digits of one in turn. For ease I'm going to treat it as AXB*CXD BD=/=X XD=/=X AD=/=X XC>1, and unless X=9, XC>3. Given as...

I recently launched a Kickstarter for a roleplaying product that's seeming quite popular: It's a deck of playing cards each of which describes an NPC so that you can quickly add them to your story when you find it's going in a direction you're not prepared for (or even when you just haven't had the ...

Very briefly, I think that consciousness is essentially a cultural construction, based on what we remember about interpreted perceptions. Nothing strange about it and certainly nothing that would, for example, separate humans from animals. This is interesting because to me consciousness isn't a thi...

I would like to refine my definition. I will explicitly accept that my definition "a capable action - An action that is theoretically possible" is not correct. I freely admit that dealing with the bounds of theoretical possibility is too unclear to be used in at least some cases, possibly...

The answer can also be calculated with my definition. It's a lot harder . Is that your problem? That it's not really really easy? Is my calculation wrong? Why does that not match reality? Because all the "theoretical" numbers are far higher than the actually achievable numbers. It's theor...

I Can calculate the answer. The answer can also be calculated with my definition. It's a lot harder . Is that your problem? That it's not really really easy? Because the answer is definitely still calculable with my definition, it's just that neither of us has the necessary information or computati...

Ok, great. It doesn't match the facts. If that's the case I will happily admit that the definition is wrong as stated, and attempt to refine it. Wrong relative to what? Itself? I can't. The real meaning? Easy, I've done so, even you've done so. I think you misquoted here... what can't you do? The r...

It's an unfortunate coincidence of language that we have some words with many meanings. It is impractical to use the explicit logical definition when communication colloquially. When I say "I can't think of any other way to say it" that's speaking colloquially, but you're right, the actua...

Holy crap these posts are getting long!!! I'll try and cut some bits where they're repetitive, see what I can manage. Those are questions of fact. Defining your meaning of "can" to avoid it being impinged on by questions of fact is a really, really bad idea. Why is that a bad idea? That s...

My definition is also unambiguous, but it's almost always the same as the common meaning. And thus your definition is a failure. My definition works fine for all three questions. I take issue here, the colloquial definition is VERY ambiguous in many cases. Whereas the logical definition is VERY cle...

To me, that's absurd. That's not an acceptable definition, because it doesn't match the normal meaning. I agree, this does not match the common or colloquial usage. When you're talking about strict logical proofs though (A --> B), a strict, unambiguous, logical definition is needed in order to main...

[snip] Yes, it is absolutely within my power and ability to go out and buy the winning lottery ticket tomorrow. Given the resources, I am capable. [snip] Yes, he is capable of answering the question... is he capable of comprehending the question? That's a different story. What do you find unreasona...

By the transitive property of equality (if a = b and b = c, then a = c) X = 1,2,3,4,5,6 X = Winning Lottery numbers "You can pick X, but you can't pick X" is nonsensical. Except that at the time of picking, you don't know that they're identical; so I'm not saying "you can pick X but ...

My failure to understand here is that this is the law of identity (or, at least, the law of reflexivity). 1,2,3,4,5,6 == Winning lottery numbers. If you pick one, that's the same thing as picking the other. Yes, but remember my definition of "can": "If you choose to, you will". ...

This is exactly what I was trying to demonstrate. There is no difference between the Opposite of the Right and the Hypotenuse. But lack of knowledge makes this person incapable picking the hypotenuse. This is the same lack of knowledge that makes a person incapable of picking the box with $1M. I ag...

My point was more that this whole idea of 2nd order Free Will seems a bit fuzzy, or even meaningless, if there's no objective way to decide what it is I really want. Kapeesh? Here's a different way to look at it: First order freedom= Doing what you want to do. (for instance, you might want to play ...

EDIT2: Sorry, I've been being a bit bitchy in this thread, and not very patient. I realise we're both just trying to get at the truth, and explain it to each other; I've been having a very frustrating time, and I've been letting it influence my behaviour. I promise to try harder to be civil. Ok, I t...

Well, yes, but that's because you can't do X. Because that's exactly how you defined it! You defined "ability" as a consequent of "choice" which it isn't. Perhaps you seem to think I'm affirming the consequent because you think that a non-consequent is a consequent. I'd have to ...

That is not a difference. That is personal ignorance leading someone to wrongly think there's a difference (or that one is unanswerable). It is the same question phrased differently. It's not that one is unanswerable, it's that the person is capable of answering one, but not of answering the other,...

I do think, however, that you're missing Trebla fallacious arguments; note how they now admits that the two statements are different, and that while mine would necessarily follow from theirs, theirs doesn't necessarily follow from mine... Now you're flat out lying. I said If you "might" (...

I've given my definition of "can" repeatedly: If they chose to do X, they would successfully do X. I've given it at least half a dozen times. Of course this includes the contentious word 'chose', which you are both using differently. But this is really between you two and I shouldn't have...

They are not the same action. Much like "Picking 1, 2, 3, 4, 5, 6" on the lottery and "picking the winning lottery numbers" are different actions , even if you later discover that 1,2,3,4,5,6 were the winning numbers. They are the same action, they are logically equivalent. &quo...

I agree they are different statements. The former is a necessary consequent of the latter. So why did you claim I was agreeing with you? Are you affirming the consequent again? Taking my agreement on might as agreement on is capable of ? If you "might" (a statement of non-zero probability...

'Can't' and 'won't' both have multiple definitions in common usage, some of which overlap. For example. It appears that you are insisting on very specific meanings that Trebla hasn't agreed to. If you were to state the precise meanings of these words as you intend them maybe this conversation could...

Doesn't follow, sorry. If you might choose differently, the prophecy can't be infallible. This is exactly what I said follows... it seems to me we're in agreement... No, no it isn't. You said, and I quote: IF you are capable of choosing differently, the prophecy is impossible to be infallible. If y...

Computers do process however. And humans do think. If you claim otherwise, explain the existence of this conversation? How am I responding to you if I don't think? How are those responses reaching you if computers can't process, can't run programs etc.? Under classical determinism the conversation ...

YES! Exactly! IF you are capable of choosing differently, the prophecy is impossible to be infallible. Doesn't follow, sorry. If you might choose differently, the prophecy can't be infallible. But you can be capable of things without any chance you'll do them (I could chop my nose off right now... ...

I perfectly understand the difference between the two. At least I think I do. If something is not possible, it cannot be done. (It also will not be done, but that's a less valuable statement). How can I cause an infallible prediction to fail? My hypothesis is that it cannot be done. Therefore, I wi...

Since the majority of your response comes down to you missing this point, I'll cut out the bulk. Given absolute predetermination, "It can be predicted that you will do X" is inviolable. You act like you're accepting predetermination, but all your examples implicitly reject it. No, they do...

You misinterpreted me, and I already admitted that the fault was in my communication. Reread, if you must (haha), the thread replacing "you must" with "it must be the case" as I was using the former as shorthand for the latter. I made the errant assumption that this was understo...